CN111176209B - Off-line planning method for feeding rate and rotating speed of cavity spiral milling - Google Patents
Off-line planning method for feeding rate and rotating speed of cavity spiral milling Download PDFInfo
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- G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
- G05B19/19—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path
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- G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
- G05B19/00—Programme-control systems
- G05B19/02—Programme-control systems electric
- G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
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- G05B19/4163—Adaptive control of feed or cutting velocity
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Abstract
The invention provides a method for offline planning of a feeding rate and a rotating speed of cavity spiral milling. First, a smooth helix of cubic spline curve is obtained. Discretizing the spline curve track equally according to parameters to obtain discrete parameter points; simulating the tool path track formed by the discrete points to obtain the radial cutting width of the discrete points; calculating a stable lobe graph, and establishing a corresponding relation between the rotating speed and the critical cutting depth; the radial cutting width and the corresponding relation between the rotating speed and the critical cutting depth at each parameter point are used for establishing a time-optimal feeding speed and rotating speed planning model at each parameter point, and the time-optimal feeding speed and rotating speed planning at each parameter point is obtained through forward planning and backward planning; fitting to obtain the speed and the rotating speed in the whole parameter interval; and finally, obtaining the position, the maximum feeding speed and the rotating speed of each tool location point, and outputting a corresponding NC code for actual machining aiming at a specific machine tool numerical control system. The method is suitable for high-speed milling of the cavity.
Description
Technical Field
The invention relates to the field of cavity, in particular to a cavity spiral milling machining feed rate and rotating speed offline planning method, and more particularly relates to a cavity spiral milling machining feed rate and rotating speed offline planning method which comprehensively meets the requirements of the kinematic performance, the mechanical performance and the stability of the machining process of a machine tool.
Background
The cavity milling is widely applied to the processing of aerospace wall plates and molds. The cavity milling is performed in a layer-by-layer processing mode, and due to the fact that the size of a part is large, long processing time is usually needed. Therefore, on the premise of ensuring high precision and high reliability, the method for machining the cavity of the die cavity has the important problems of obviously improving the machining efficiency and reducing the time required by machining.
At present, a commonly used feed rate optimization algorithm for cavity spiral milling is usually based on kinematic performance constraints of a machine tool, such as speed and acceleration, and influences of mechanical properties and stability of a machining process are not considered. If the cutting force in the machining process is too large, the cutter can be damaged, and the service life of the cutter is influenced; if flutter occurs in the machining process, the surface of a workpiece is corrugated, the machining quality and precision are reduced, even the machine tool and a cutter are damaged in serious cases, the machining cost is increased, and the machining efficiency is reduced.
The invention patent with publication number 108145222B discloses a milling method for a closed blisk cavity, which comprises the following steps: dividing a rough machining area of the cavity; step 2: determining to-be-processed area division; and step 3: planning a tool path; the milling method for the closed blisk cavity can accurately divide blocks, reduces machining allowance, is simple in cutter shaft vector solving, realizes five-axis numerical control machining of the efficient closed blisk cavity, improves machining quality and machining efficiency of the closed blisk, can be applied to milling machining of blisk blades and can also be applied to machining processes of other thin-wall large-overhang parts. However, the above patent has a problem of low processing efficiency.
Disclosure of Invention
Aiming at the defects in the prior art, the invention aims to provide an off-line programming method for the feeding rate and the rotating speed of cavity spiral milling.
The invention provides a method for offline planning of the feed rate and the rotating speed of cavity spiral milling, which comprises the following steps:
step 1: generating a smooth spiral line track in a cubic B-spline curve form according to the shape of the cavity and the diameter of the cutter;
and 2, step: carrying out equal-parameter discrete sampling on the spiral track to obtain discrete parameter points;
and step 3: obtaining the radial cutting width at each discrete parameter point through VERICUT simulation;
and 4, step 4: calculating a stable lobe graph by utilizing the maximum radial cutting width obtained by simulation, and establishing a corresponding relation between the rotating speed and the critical cutting depth;
and 5: forward planning, namely establishing a time-optimal feeding speed and rotating speed planning model for each parameter point by taking the motion speed of a machine tool, the acceleration of each axis, the height error of a curve, the maximum cutting force and the machining stability as constraints in the direction of increasing the parameter value, and solving the model by using a differential evolution algorithm to calculate the maximum feeding speed and the corresponding rotating speed at each discrete parameter point;
step 6: backward planning, namely establishing and solving a planning model of optimal time feeding speed and rotating speed along the direction of reducing the parameter values;
and 7: and performing fitting approximation on the speed and the rotating speed in the whole parameter interval, and utilizing spline curve interpolation to obtain the feeding speed and the rotating speed at each sampling point, so as to obtain the position, the maximum feeding speed and the rotating speed of each tool location point, and finally outputting the NC codes as the specific NC codes of the machine tool numerical control system.
Preferably, the step 1 comprises:
the method comprises the steps of solving an elliptic partial differential equation of the Dirichlet boundary condition by adopting a finite element method, generating a closed field curve with a closed equivalent according to the diameter of a cutter, generating a transition curve after equally dividing the field curve to obtain a group of spiral point rows, and fitting the spiral point rows by utilizing a cubic B-spline curve to obtain a cubic B-spline curve form smooth spiral line track.
Preferably, the step 2 includes:
and (3) equally dividing the whole parameter interval into N points, wherein u1 is 0, uN is 1, and substituting the three-time B-spline curve to obtain each discrete parameter point.
Preferably, the step 3 comprises:
and (3) taking the discrete parameter points obtained in the step (2) as tool positions, setting initial fixed rotating speed and fixed feed rate to generate initial NC codes, importing the codes into VERICUT for simulation, and obtaining the radial cutting width of each tool position after the simulation is finished so as to obtain the radial cutting width of each discrete point.
Preferably, the step 5 comprises:
let the cubic B-spline curve form smooth spiral trajectory be C (u), let κ (u) | survival of the eyesC u (u) | | represents the speed of the parameter, where () u Represents the derivative of the variable with respect to u, V (u) represents the feed rate;
the time-optimal objective function is
Wherein s is the chord length, v is the magnitude of the velocity, and u' is the derivative of the parameter u with respect to time;
the time-optimal objective function can be further expressed as:
wherein N is the number of sampling points;
setting the speed of the head and tail points to be zero, only establishing a time-optimal feed speed and rotating speed planning model for each discrete parameter point of i-2, 3, … and N-1, wherein the objective function is
While
Then can obtain
Stipulate () μ And (mu epsilon { x, y }) is a component of the vector in the mu direction, and the method for establishing the constraint of the acceleration of each axis of the machine tool comprises the following steps:
wherein () u Representing the derivative of the variable with respect to u, with V μ representing the speed of the machine axes, A μ Indicating acceleration of machine tool axesAnd (4) degree. The curve bow height error constraint can be expressed as
Wherein delta m For a given bow height error, T s For the interpolation period, ρ (u) is the radius of curvature of the spline curve at parameter u, and
dispersing the equations (4) and (5) as follows to obtain the limit constraint inequality of the movement speed and the acceleration of each axis of the machine tool
V in formula (7) max And
respectively representing the feed speed limit of the machine tool and the acceleration limit of each shaft;
calculating the cutting force at each discrete parameter point in the cutting process according to the curvature and the radial cutting depth at each discrete parameter point; the cutting depth part of the milling cutter is axially dispersed into a plurality of layers, and the cutting force of the cutting edge infinitesimal is as follows:
wherein j is the cutter tooth number, and k is the cutter shaft direction disc number; dF t,j,k The unit tangential cutting force of the jth cutter tooth on the kth disc; dF r,j,k The unit radial cutting force of the jth cutter tooth on the kth disc; h is j,k For non-deformation thickness cutting, the feed rate per tooth of the cutter is f t A function of (a); k te And K re Is the edge cutting force coefficient, K, of the tool tc And K rc Is a shearing cutting forceA coefficient; phi is a j,k (t) is the contact angle of the jth cutter tooth on the kth disc, and the unit step function g (phi) j,k (t)) is used for indicating whether the current cutting element participates in cutting, and is defined as follows:
wherein phi st And phi ex The cutting angle and the cutting angle of the jth cutter tooth on the kth disc are represented by the following calculation formula:
wherein R is the curvature radius at the discrete parameter point, R is the cutter radius, and a is the radial cutting width at the discrete parameter point; k te 、K re 、K tc And K rc The cutting force coefficient can be obtained by milling a workpiece with a simple shape and the same material by using the same milling cutter, designing a cutting experiment of a specific processing parameter, recording corresponding cutting force data by using a dynamometer and finally calibrating by using a least square method;
the infinitesimal cutting force determined by the formula (8) is transformed into a tool coordinate system through coordinate transformation:
the instantaneous resultant forces acting in the two directions transverse to the tool are:
wherein N is t Number of teeth of knife, N A The number of the disks is axially discrete;
obtaining the maximum value of resultant force borne by the cutter at each discrete parameter point by the formula (8-12), wherein the maximum resultant force F is the feed rate F of each tooth t Is composed ofNumber, and feed rate per tooth f t The relationship between the feeding speed V and the rotating speed omega is as follows:
wherein N is t The number of the cutter teeth;
the maximum resultant force F at each discrete parameter point is thus a function of the feed speed V and the rotational speed omega, expressed as
The cutting force constraint is expressed as:
wherein F max The cutting force threshold is determined by the strength of the cutter teeth, the integral rigidity of the cutter and the maximum deformation of the cutter;
the correspondence of the rotational speed to the critical depth of cut has been established by step (4), and is denoted b lim (omega), the layer-by-layer processing of the cavity can set a certain cutting depth value b set Taking this value as the lower threshold of the critical cutting depth, the cutting force constraint is expressed as:
b lim (Ω(u i ))≥b set (15)
converting the nonlinear constraints of the equations (14) and (15) into penalty terms by an external penalty function method, and integrating the penalty terms into an objective function
Wherein F max As cutting force threshold, b set For the set layer-by-layer machining cutting depth value, a time optimal feed speed and rotating speed planning model comprising machine tool speed, acceleration of each axis, curve bow height error constraint, maximum cutting force constraint and machining process stability constraint can be constructed:
where σ is a penalty factor, P (V (u) i ),Ω(u i ) σ P (V (u) as defined in formula (16) i ),Ω(u i ) V) as a penalty including maximum cutting force constraint, process stability constraint max And
respectively the machine tool feed speed limit and the acceleration limit, omega, of each axis min And Ω max Is a defined range of rotational speeds;
since the cutting force constraint and the stability constraint in the equations (14) and (15) are both nonlinear constraints, the model is solved through a differential evolution algorithm; continuously solving and calculating along the direction of increasing parameter values to obtain the maximum feeding speed and the corresponding rotating speed at each discrete parameter point, and recording as { V } i * ,Ω i * ,i＝2,…,N-1}。
Preferably, the step 6 includes:
and continuously solving and calculating the maximum feeding speed and the corresponding rotating speed at each discrete parameter point by adopting a differential evolution algorithm along the direction of reducing the parameter values, wherein the calculated values are the final time optimal feeding speed and rotating speed planning result.
Compared with the prior art, the invention has the following beneficial effects:
1. the invention provides a method for offline planning of a feeding rate and a rotating speed of cavity spiral milling, which meets the following 4 constraints: 1) constraining the motion speed of the machine tool and the acceleration of each shaft; 2) constraining the height error of the curve; 3) maximum cutting force constraint; 4) and (5) stability constraint of the processing process.
2. The method is suitable for high-speed milling of the cavity.
Drawings
Other features, objects and advantages of the invention will become more apparent upon reading of the detailed description of non-limiting embodiments with reference to the following drawings:
FIG. 1 is a schematic flow diagram of the process of the present invention.
Fig. 2 is a schematic diagram of a path of a rectangular cavity spiral milling cutter.
FIG. 3 is a graph of a stable lobe calculated using the maximum radial cut width.
FIG. 4 shows a flat-bottomed cylindrical milling cutter with a partial dispersion of the cutting depth in the axial direction of N A And the disk units with the same height.
Fig. 5 is a schematic view of a machining engagement area.
The figures show that:
1 is a cavity boundary, 2 is a spiral track outer boundary, 3 is a spiral track inner boundary, 4 is a smooth spiral line track in a cubic B-spline curve form, 5 is a flat-bottom cylindrical milling cutter, 6 is a cutting edge of a cutter, and 7 is a cutter path track;
omega is the rotational speed, b lim Is the critical depth of cut, R is the radius of curvature at the discrete parameter point, R is the radius of the tool, a is the radial cutting width at the discrete parameter point, φ st XOY is the tool coordinate system and Δ z is the height of the discrete disk element for the entry angle of the engagement zone.
Detailed Description
The present invention will be described in detail with reference to specific examples. The following examples will assist those skilled in the art in further understanding the invention, but are not intended to limit the invention in any way. It should be noted that it would be obvious to those skilled in the art that various changes and modifications can be made without departing from the spirit of the invention. All falling within the scope of the present invention.
As shown in fig. 1 to 5, the present invention provides a method for planning the feeding speed and the rotation speed with the time optimization as the target, that is, a feeding speed and rotation speed planning model with the constraints of the machine tool motion speed, the acceleration of each motion axis and the curve height error as the constraints is established at each discrete parameter point, and with the constraints of the maximum cutting force in the cutting process of the tool being smaller than the threshold value and the stability constraint of the critical depth of cut corresponding to the rotation speed being larger than the specified layer-by-layer cutting depth is established, so as to maximize the feeding speed at each parameter point. Because the acceleration at the parameter point is influenced by the speed of the adjacent parameter points, the parameter points are respectively planned in a forward process and a backward process, and finally, the optimal planning result is obtained. After the optimal speed and the optimal rotating speed of each parameter point are obtained, fitting approximation is carried out on the speed and the rotating speed in the whole parameter interval, the feeding speed and the rotating speed of each sampling point are obtained by spline curve interpolation, then the maximum feeding speed and the corresponding rotating speed of each tool position can be obtained, and finally the NC codes of the specific machine tool numerical control system are output.
The method specifically comprises the following steps:
1) and generating a smooth spiral line track in a cubic B-spline curve form according to the shape of the cavity and the diameter of the cutter.
2) And carrying out equal-parameter discrete sampling on the spline curve track to obtain each discrete parameter point.
3) And obtaining the radial cutting width at each discrete parameter point through VERICUT simulation.
4) And calculating a stable lobe graph by utilizing the maximum radial cutting width obtained by simulation, and establishing a corresponding relation between the rotating speed and the critical cutting depth.
5) And (6) planning in a forward direction. And (3) along the direction of increasing the parameter values, sequentially establishing a time-optimal feed speed and rotating speed planning model for each parameter point by taking the motion speed of the machine tool, the acceleration of each axis, the curve height error, the maximum cutting force and the machining stability as constraints. And solving the model by using a differential evolution algorithm to calculate the maximum feeding speed and the corresponding rotating speed at each discrete parameter point.
The cutting force threshold is determined by the strength of the cutter teeth, the integral rigidity of the cutter and the maximum deformation of the cutter.
And a certain cutting depth value is set for layer-by-layer processing of the cavity, and the value is used as a lower threshold of the critical cutting depth, namely, the stability of the processing process is ensured by the fact that the critical cutting depth corresponding to the rotating speed is larger than the lower threshold.
6) And (5) backward planning. And (4) establishing and solving a time-optimal feed speed and rotating speed planning model along the direction of reducing the parameter value.
The method of modeling along the direction of decreasing parameter values is substantially the same as step 5, and the maximum feeding speed of each parameter point obtained in step 5 is increased to be the upper limit of the speed of each parameter point.
7) And performing fitting approximation on the speed and the rotating speed in the whole parameter interval, and utilizing spline curve interpolation to obtain the feeding speed and the rotating speed at each sampling point, so as to obtain the position, the maximum feeding speed and the rotating speed of each tool location point, and finally outputting the NC codes of the specific machine tool numerical control system.
In the following embodiments, taking the rectangular cavity processing (fig. 2) as an example, a similar method can be applied to any cavity with a complicated single connected region shape and a single island. And the cutter is set to be a down-milling cutter.
1) And generating a smooth spiral line track in a cubic B-spline curve form according to the shape of the cavity and the diameter of the cutter.
Before the cavity is processed by spiral milling, spiral cutting needs to be carried out by taking a value which is not more than the radius of a cutter as a spiral radius, so that the inner boundary of a coverage area of a spiral milling cutter path is a circle obtained by projecting a spiral cutting path on a plane; in order to prevent over-cutting of the cavity boundary, the cavity shape needs to be biased inward by the radius of the cutter to obtain the outer boundary of the coverage area of the spiral milling cutter path.
Solving an elliptic partial differential equation of the Dirichlet boundary condition by adopting a finite element method, generating a closed equivalent field curve according to the diameter of a cutter, generating a transition curve after equally dividing the field curve to obtain a group of spiral point lines, and fitting the spiral point lines by utilizing a cubic B spline curve to obtain a cubic B spline curve form smooth spiral line track.
A specific generation process of a helical milling Tool Path in the form of a spline curve can be found in Bieterman M B, Sandstrom D R.A Curvilinerol-Path Method for Pocket Machining [ J ]. Journal of Manufacturing Science and Engineering,2003,125(4):709 ]
2) And carrying out equal-parameter discrete sampling on the spline curve track to obtain each discrete parameter point.
Changing u to [0,1]]The whole parameter interval is equally divided into N points, wherein u 1 ＝0，u N 1, generationAnd putting the cubic B-spline curve to obtain each discrete parameter point.
3) The radial cutting width at each discrete point was obtained by vericu simulation.
And (3) taking the discrete parameter points obtained in the step (2) as tool location points, setting an initial fixed rotating speed and a fixed feed rate to generate a preliminary NC code, importing the preliminary NC code into VERICUT for simulation, and obtaining the radial cutting width of each tool location point after the simulation is finished so as to obtain the radial cutting width of each discrete point.
4) And calculating a stable lobe graph by utilizing the maximum radial cutting width obtained by simulation, and establishing a corresponding relation between the rotating speed and the critical cutting depth.
Because the larger the radial cutting width is, the smaller the corresponding critical cutting depth is at the same rotating speed, and if the processing at the point with the maximum radial cutting width is stable, the stability of the whole cavity processing process can be ensured. Therefore, the maximum value of the radial cutting width at each discrete point obtained in the step (3) is selected as the radial cutting width to establish a stable lobe graph, as shown in fig. 3.
The stable lobe plot may be created using either a single frequency method, which may be referenced in the literature [ Altintas Y, Budak E.analytical Prediction of Stability Lobes in Milling [ J ]. CIRP Annals-Manufacturing Technology,1995,44(1):357 @ ], or a full dispersion method, which may be referenced in the literature [ Ding Y, Zhu L M, Zhuang X J, et al.A full-dispersion method for Prediction of compliance [ J ]. International Journal of Machine Tools & Manufacturing, 2010,50(5): 502-. The used tool modal parameters can be obtained through a hammering test, the cutting force coefficient can be obtained through milling a workpiece (such as a rectangular workpiece end mill) with a simple shape and the same material by using the same milling cutter, designing a cutting experiment (such as slotting) with specific processing parameters, recording corresponding cutting force data by using a dynamometer, and finally calibrating by using a least square method.
5) And (6) planning in a forward direction. And (3) along the direction of increasing the parameter values, sequentially establishing a time-optimal feed speed and rotating speed planning model for each parameter point by taking the speed, the acceleration, the curve height error, the maximum cutting force and the processing stability of each axis of the machine tool as constraints. And solving the model by using a differential evolution algorithm to calculate the maximum feeding speed and the corresponding rotating speed at each discrete parameter point.
Let C (u) be a cubic B-spline curve type smooth spiral line track, and let κ (u) | C u (u) | | represents the parameter velocity, where () u Represents the derivative of the variable with respect to u, and v (u) represents the feed rate. The time-optimal objective function is
Where s is the chord length, v is the magnitude of the velocity, and u' is the derivative of the parameter u with respect to time.
It is difficult to directly solve the formula (1), and isoparametric discrete sampling can be carried out on the parameter interval of the motion curve of the machine tool, so that the time-optimal objective function can be expressed as
Where N is the number of sample points.
Setting the speed of the head and tail points to be zero, only establishing a time-optimal feed speed and rotating speed planning model for each discrete parameter point of i-2, 3, … and N-1, wherein the objective function is
While
Then can obtain
Stipulated () μ (mu e { x, y }) is the component of the vector in the mu direction, and the constraint establishment method of the acceleration of each axis of the machine tool is as follows
Wherein () u Representing the derivative of the variable with respect to u, V μ Indicating the speed of each axis of the machine tool, A μ Representing the acceleration of each axis of the machine tool. The curve height error constraint can be expressed as
Wherein delta m For a given bow height error, T s For the interpolation period, ρ (u) is the radius of curvature of the spline curve at parameter u, and
solving for V (u) in view of the forward planning process i ) While the parameter is decreased toward the feeding speed V (u) of the adjacent discrete parameter point i-1 ) Knowing, however, the feed speed V (u) of the discrete parameter points adjacent in the direction of increasing parameter i+1 ) Unknown, therefore, the equations (4) and (5) are discretized into the constraint inequalities of the motion speed and the acceleration limit of each axis
V in formula (6) max And
respectively the machine tool feed speed limit and the acceleration limit of each axis. The derivation of the above formula can be found in the literature [ Dong W, Ding Y, Huang J, et al].Journal of Dynamic Systems,Measurement,and Control,2017,139(6):061012.].
And calculating the cutting force at each discrete parameter point in the cutting process according to the curvature and the radial cutting depth at each discrete parameter point. The milling cutter cutting depth portion is axially discretized into several layers as shown in fig. 4. The cutting edge infinitesimal cutting force can be expressed as:
wherein j is the cutter tooth number, and k is the cutter shaft direction disc number; dF t,j,k The unit tangential cutting force of the jth cutter tooth on the kth disc; dF r,j,k The unit radial cutting force of the jth cutter tooth on the kth disc; h is a total of j,k For non-deformation thickness cutting, the feed rate per tooth of the cutter is f t A function of (a); k te And K re Is the edge cutting force coefficient, K, of the tool tc And K rc Is the shear cutting force coefficient. Phi is a j,k And (t) is the contact angle of the jth cutter tooth on the kth disc. Unit step function g (phi) j,k (t)) is used for indicating whether the current cutting element participates in cutting, and is defined as follows:
wherein phi st And phi ex The cutting angle and the cutting angle of the jth cutter tooth on the kth disc are represented by the following calculation formula:
where R is the radius of curvature at the discrete parameter point, R is the tool radius, and a is the radial cut width at the discrete parameter point, as shown in fig. 5. The derivation process can be referred to as [ Zhang, l., Zheng, l., Zhang, z. -h., Liu, y.,&Li,Z.-Z. (2002).On cutting forces in peripheral milling of curved surfaces.Proceedings of the Institution of Mechanical Engineers,Part B:Journal of Engineering Manufacture,216(10), 1385–1398.]。K te 、K re 、K tc and K rc Equal cuttingThe force coefficient can be obtained by milling a workpiece with a simple shape (such as a rectangular workpiece) made of the same material by using the same milling cutter, designing a cutting experiment (such as slotting) with specific processing parameters, recording corresponding cutting force data by using a dynamometer, and finally calibrating by using a least square method.
The infinitesimal cutting force determined by the formula (8) is transformed into a tool coordinate system through coordinate transformation.
The instantaneous resultant forces acting in the two directions transverse to the tool are:
wherein N is t Number of teeth of knife, N A The number of axially discrete discs of the tool.
The maximum value of the resultant force on the cutter at each discrete parameter point can be obtained by the formula (8-12). The maximum resultant force F is the feed rate per tooth F according to the calculation process t As a function of, and the feed rate per tooth f t The relationship between the feeding speed V and the rotating speed omega is as follows:
wherein N is t The number of teeth is the number of teeth.
The maximum resultant force F at each discrete parameter point is thus a function of the feed speed V and the rotational speed Ω, which can be expressed as
The cutting force constraint can therefore be expressed as:
wherein F max Is the cutting force threshold determined by the strength of the cutter teeth, the overall stiffness of the cutter and the maximum deformation of the cutter.
The correspondence of the rotation speed to the critical depth of cut has been established by step (4), and this relationship is denoted b lim (omega), the layer-by-layer processing of the cavity can set a certain cutting depth value b set And taking the value as a lower threshold of the critical cutting depth, namely ensuring the stability of the machining process by using the critical cutting depth corresponding to the rotating speed to be larger than the lower threshold. The cutting force constraint can therefore be expressed as:
b lim (Ω(u i ))≥b set (15)
converting the nonlinear constraints of the equations (14) and (15) into penalty terms integrated into an objective function by an external penalty function method, and enabling the penalty terms to be integrated into the objective function
Wherein F max As cutting force threshold, b set For the set layer-by-layer machining cutting depth value, a time optimal feed speed and rotating speed planning model comprising machine tool speed, acceleration of each axis, curve bow height error constraint, maximum cutting force constraint and machining process stability constraint can be constructed:
where σ is a penalty factor (a very large positive number), P (V (u) i ),Ω(u i ) σ P (V (u) as defined in formula (16) i ),Ω(u i ) V) as a penalty including maximum cutting force constraint, process stability constraint max And
respectively the machine tool feed speed limit and the acceleration limit, omega, of each axis min And Ω max Is a defined range of rotational speeds.
Since the cutting force constraint and the stability constraint in equations (14), (15) are both nonlinear constraints, the model can be solved by a differential evolution algorithm. Continuously solving and calculating along the direction of increasing parameter values to obtain the maximum feeding speed and the corresponding rotating speed at each discrete parameter point, and recording as { V } i * ,Ω i * ,i＝2,…,N-1}。
6) And (5) backward planning. And establishing and solving a time-optimal feed speed and rotating speed planning model along the direction of reducing the parameter value.
Because the acceleration condition between the N-1 th parameter point and the Nth parameter point cannot be considered in the forward planning process, the feed speed and the rotating speed between the N-1 st parameter point and the 2 nd parameter point need to be corrected by adding the backward planning process.
The process of building the backward planning model is generally consistent with, but different from, forward planning. Firstly, the maximum feed speed V obtained in the step 5 needs to be increased i * For the upper speed bound of each parameter point, additionally solving V (u) in the backward planning process i ) While the parameter is increased toward the feeding speed V (u) of the adjacent discrete parameter point i+1 ) Knowing, and decreasing the feed velocity V (u) of the discrete parameter points adjacent thereto i-1 ) Unknown and thus the discrete mode of equation (4) differs.
The time optimal feed speed and rotating speed planning model which is constructed after adjustment and contains machine tool speed, acceleration of each axis, curve bow height error constraint, maximum cutting force constraint and machining process stability constraint is as follows:
where σ is a penalty factor (a very large positive number), P (V (u) i ),Ω(u i ) σ P (V (u) as defined in formula (16) i ),Ω(u i ) V) as a penalty including maximum cutting force constraint, process stability constraint max And
respectively the machine tool feed speed limit and the acceleration limit of each axis, omega min And Ω max Is a defined range of rotational speeds.
And (5) continuously solving and calculating the maximum feeding speed and the corresponding rotating speed at each discrete parameter point by adopting a differential evolution algorithm along the direction of reducing the parameter value, wherein the calculated value is the final time optimal feeding speed and rotating speed planning result.
7) And performing fitting approximation on the speed and the rotating speed in the whole parameter interval, and utilizing spline curve interpolation to obtain the feeding speed and the rotating speed at each sampling point, so as to obtain the position, the maximum feeding speed and the rotating speed of each tool location point, and finally outputting the NC codes of the specific machine tool numerical control system.
The foregoing description of specific embodiments of the present invention has been presented. It is to be understood that the present invention is not limited to the specific embodiments described above, and that various changes or modifications may be made by one skilled in the art within the scope of the appended claims without affecting the spirit of the invention. The embodiments and features of the embodiments of the present application may be combined with each other arbitrarily without conflict.
Claims (6)
1. The off-line planning method for the feed rate and the rotating speed of cavity spiral milling machining is characterized by comprising the following steps of:
step 1: generating a cubic B-spline curve-form smooth spiral line track according to the shape of the cavity and the diameter of the cutter;
step 2: carrying out equal-parameter discrete sampling on the spiral track to obtain discrete parameter points;
and step 3: obtaining the radial cutting width of each discrete parameter point through simulation;
and 4, step 4: calculating a stable lobe graph by utilizing the maximum radial cutting width obtained by simulation, and establishing a corresponding relation between the rotating speed and the critical cutting depth;
and 5: forward planning, namely establishing a time-optimal feeding speed and rotating speed planning model along the direction of increasing the parameter values, and solving to obtain the maximum feeding speed and the corresponding rotating speed at each discrete parameter point;
step 6: backward planning, namely establishing a time-optimal feeding speed and rotating speed planning model along the direction of reducing the parameter values, and solving to obtain the maximum feeding speed and the corresponding rotating speed at each discrete parameter point;
and 7: fitting approximation is carried out on the maximum feeding speed and the rotating speed in the whole parameter interval, the feeding speed and the rotating speed at each sampling point are obtained by utilizing spline curve interpolation, and then the position, the maximum feeding speed and the rotating speed of each cutter point can be obtained.
2. The off-line cavity spiral milling machining feed rate and rotation speed planning method according to claim 1, wherein the step 1 comprises the following steps:
the method comprises the steps of solving an elliptic partial differential equation of the Dirichlet boundary condition by adopting a finite element method, generating a closed field curve with a closed equivalent according to the diameter of a cutter, generating a transition curve after equally dividing the field curve to obtain a group of spiral point rows, and fitting the spiral point rows by utilizing a cubic B-spline curve to obtain a cubic B-spline curve form smooth spiral line track.
3. The off-line cavity spiral milling machining feed rate and rotation speed planning method according to claim 1, wherein the step 2 comprises:
and (3) dividing the whole parameter interval into N points equally, wherein u1 is 0, uN is 1, and substituting the points into a cubic B spline curve to obtain each discrete parameter point.
4. The off-line cavity spiral milling machining feed rate and rotation speed planning method according to claim 1, wherein the step 3 comprises:
and (3) taking the discrete parameter points obtained in the step (2) as tool location points, setting an initial fixed rotating speed and a fixed feed rate to generate a preliminary NC code, importing the preliminary NC code into VERICUT for simulation, and obtaining the radial cutting width of each tool location point after the simulation is finished so as to obtain the radial cutting width of each discrete point.
5. The off-line cavity spiral milling machining feed rate and rotation speed planning method according to claim 1, wherein the step 5 comprises:
let the cubic B-spline curve form smooth spiral trajectory be C (u), let κ (u) | | C u (u) | | represents the speed of the parameter, where () u Represents the derivative of the variable with respect to u, V (u) represents the feed rate;
the time-optimal objective function is
Wherein s is the chord length, v is the magnitude of the velocity, and u' is the derivative of the parameter u with respect to time;
the time-optimal objective function can be further expressed as:
wherein N is the number of sampling points;
setting the speed of the head and tail points to be zero, only establishing a time-optimal feed speed and rotation speed planning model for each discrete parameter point of i-2, 3, …, N-1, wherein the objective function is
While
Then can obtain
Stipulated () μ (mu e { x, y }) is the component of the vector in the mu direction, and the constraint establishment method of the acceleration of each axis of the machine tool is as follows:
wherein () u Representing the derivative of the variable with respect to u, V μ Indicating the speed of each axis of the machine tool, A μ Representing the acceleration of each axis of the machine tool, and the curve height error constraint can be expressed as
Wherein delta m For a given bow height error, T s For the interpolation period, ρ (u) is the radius of curvature of the spline curve at parameter u, and
dispersing the equations (4) and (5) as follows to obtain the limit constraint inequality of the movement speed and the acceleration of each axis of the machine tool
V in formula (7) max And with
Respectively representing the feed speed limit of the machine tool and the acceleration limit of each shaft;
calculating the cutting force at each discrete parameter point in the cutting process according to the curvature and the radial cutting depth at each discrete parameter point; the cutting depth part of the milling cutter is axially dispersed into a plurality of layers, and the cutting force of the cutting edge infinitesimal is as follows:
wherein j is the cutter tooth number, and k is the cutter shaft direction disc number; dF t,j,k The unit tangential cutting force of the jth cutter tooth on the kth disc; dF r,j,k The unit radial cutting force of the jth cutter tooth on the kth disc; h is j,k For non-deformed thickness cutting, the feed rate per tooth of the cutter is f t A function of (a); k te And K re Is the edge cutting force coefficient, K, of the tool tc And K rc Is the shear cutting force coefficient; phi is a j,k (t) is the contact angle of the jth cutter tooth on the kth disc, and the unit step function g (phi) j,k (t)) is used for indicating whether the current cutting element participates in cutting, and is defined as:
wherein phi st And phi ex The cutting angle and the cutting angle of the jth cutter tooth on the kth disc are represented by the following calculation formula:
wherein R is the curvature radius at the discrete parameter point, R is the radius of the cutter, and a is the radial cutting width at the discrete parameter point; k te 、K re 、K tc And K rc The cutting force coefficient can be obtained by milling a workpiece with a simple shape and the same material by using the same milling cutter, designing a cutting experiment with specific processing parameters, recording corresponding cutting force data by using a dynamometer, and finally calibrating by using a least square method;
the infinitesimal cutting force determined by the formula (8) is transformed into a tool coordinate system through coordinate transformation:
the instantaneous resultant forces acting in the two directions transverse to the tool are:
wherein N is t Number of teeth of the knife, N A The number of the disks is axially discrete;
obtaining the maximum value of resultant force borne by the cutter at each discrete parameter point by the formula (8-12), wherein the maximum resultant force F is the feed rate F of each tooth t As a function of, and the feed rate per tooth f t The relationship between the feeding speed V and the rotation speed Ω is:
wherein N is t The number of the cutter teeth;
the maximum resultant force F at each discrete parameter point is thus a function of the feed speed V and the rotational speed omega, expressed as
The cutting force constraint is expressed as:
wherein F max The cutting force threshold is determined by the strength of the cutter teeth, the integral rigidity of the cutter and the maximum deformation of the cutter;
the correspondence of the rotational speed to the critical depth of cut has been established by step (4), and is denoted b lim (omega), the layer-by-layer processing of the cavity can set a certain cutting depth value b set Taking this value as the lower threshold of the critical cutting depth, the cutting force constraint is expressed as:
b lim (Ω(u i ))≥b set (15)
converting the nonlinear constraints of the equations (14) and (15) into penalty terms by an external penalty function method, and integrating the penalty terms into an objective function
Wherein F max As cutting force threshold, b set For the set layer-by-layer machining cutting depth value, a time optimal feed speed and rotating speed planning model comprising machine tool speed, acceleration of each axis, curve bow height error constraint, maximum cutting force constraint and machining process stability constraint can be constructed:
where σ is a penalty factor, P (V (u) i ),Ω(u i ) σ P (V (u) as defined in formula (16) i ),Ω(u i ) V) as a penalty including maximum cutting force constraint, process stability constraint max And with
Respectively the machine tool feed speed limit and the acceleration limit of each axis, omega min And Ω max Is a defined range of rotational speeds;
since the cutting force constraint and the stability constraint in the equations (14) and (15) are both nonlinear constraints, the model is solved through a differential evolution algorithm; continuously solving and calculating along the direction of increasing parameter values to obtain the maximum feeding speed and the corresponding rotating speed at each discrete parameter point, and recording as { V } i * ,Ω i * ,i＝2,…,N-1}。
6. The off-line cavity spiral milling machining feed rate and rotation speed planning method according to claim 1, wherein the step 6 comprises:
and continuously solving and calculating the maximum feeding speed and the corresponding rotating speed at each discrete parameter point by adopting a differential evolution algorithm along the direction of reducing the parameter values, wherein the calculated values are the final time optimal feeding speed and rotating speed planning result.
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